Problem: What is the product of $\sqrt[4]{25}$ and $\sqrt[6]{125}$?
We have $\sqrt[4]{25} \cdot \sqrt[6]{125} = 25^{\frac{1}{4}}\cdot 125^{\frac{1}{6}} = (5^2)^{\frac{1}{4}}\cdot (5^3)^{\frac{1}{6}}=5^{2\cdot \frac{1}{4}}\cdot 5^{3\cdot \frac{1}{6}} = 5^{\frac{1}{2}}\cdot 5^{\frac{1}{2}}=5^{\frac{1}{2}+\frac{1}{2}}=5^1=\boxed{5}$.